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Equivalence of Lp Stability and Exponential Stability of Nonlinear Lipschitzian Semigroups

Published online by Cambridge University Press:  20 November 2018

Song Xueli  and
Peng Jigen
Song Xueli
Affiliation:
Department of Mathematics and Information Science, Chang’an University, Xi’an 710064, ChinaandDepartment of Mathematics, Xi’an Jiaotong University, Xi’an 710049, Chinae-mail: songxl810@sohu.com
Peng Jigen
Affiliation:
Department of Mathematics, Xi’an Jiaotong University, Xi’an 710049, Chinae-mail: jgpeng@mail.xjtu.edu.cn
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Abstract

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${{L}_{p}}$ stability and exponential stability are two important concepts for nonlinear dynamic systems. In this paper, we prove that a nonlinear exponentially bounded Lipschitzian semigroup is exponentially stable if and only if the semigroup is ${{L}_{p}}$ stable for some $p\,>\,0$. Based on the equivalence, we derive two sufficient conditions for exponential stability of the nonlinear semigroup. The results obtained extend and improve some existing ones.

Keywords

34D0547H20exponentially stableLp stablenonlinear Lipschitzian semigroups
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 55 , Issue 4 , 01 December 2012 , pp. 882 - 889
Copyright
Copyright © Canadian Mathematical Society 2012

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